The Annals of Probability

Critical random hypergraphs: The emergence of a giant set of identifiable vertices

Christina Goldschmidt

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We consider a model for random hypergraphs with identifiability, an analogue of connectedness. This model has a phase transition in the proportion of identifiable vertices when the underlying random graph becomes critical. The phase transition takes various forms, depending on the values of the parameters controlling the different types of hyperedges. It may be continuous as in a random graph. (In fact, when there are no higher-order edges, it is exactly the emergence of the giant component.) In this case, there is a sequence of possible sizes of “components” (including but not restricted to N2/3). Alternatively, the phase transition may be discontinuous. We are particularly interested in the nature of the discontinuous phase transition and are able to exhibit precise asymptotics. Our method extends a result of Aldous [Ann. Probab. 25 (1997) 812–854] on component sizes in a random graph.

Article information

Ann. Probab. Volume 33, Number 4 (2005), 1573-1600.

First available in Project Euclid: 1 July 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 05C65: Hypergraphs 05C80: Random graphs [See also 60B20] 60F17: Functional limit theorems; invariance principles

Random hypergraphs giant component phase transition identifiability


Goldschmidt, Christina. Critical random hypergraphs: The emergence of a giant set of identifiable vertices. Ann. Probab. 33 (2005), no. 4, 1573--1600. doi:10.1214/009117904000000847.

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